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Distinct Volume Subsets Via Indiscernibles

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Distinct Volume Subsets Via Indiscernibles Distinct Volume Subsets via Indiscernibles William Gasarch ∗ Douglas Ulrich † July 19, 2018 Abstract Erd˝os proved that for every infinite X ⊆ Rd there is Y ⊆ X with |Y | = |X|, such that all pairs of points from Y have distinct distances, and he gave partial results for general a-ary volume. In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose T is a stable theory, ∆ is a finite set of formulas of T , M |= T , and X is an infinite subset of M. Then there is Y ⊆ X with |Y | = |X| and an equivalence relation E on Y with infinitely many classes, each class infinite, such that Y is (∆, E)-indiscernible. We also consider the definable version of these problems, for example we assume X ⊆ Rd is perfect (in the topological sense) and we find some perfect Y ⊆ X with all distances distinct. Finally we show that Erd˝os’s theorem requires some use of the axiom of choice. 1 Introduction In this paper we use the term 1-ary volume for length, 2-ary volume for area, 3-ary volume for volume. We may use the term volume when the dimension is understood. Also, the natural number n is identified with the set {0,...,n − 1}. A set X ⊆ Rd is a-rainbow if all a-sets of points that yield nonzero volumes have distinct d volumes. Let ha,d(n) be the largest integer t such that any set of n points in R contains a rainbow subset of t. This function was studied by Conlon et. al [2] which also includes references to past work. arXiv:1807.06654v1 [math.LO] 17 Jul 2018 In this paper, we are interested in the case where the cardinality of the set of points is some ℵ0 κ with ℵ0 ≤ κ ≤ 2 . Erd˝os was the first to consider this in [3]. Using different terminology, he proved the following: Theorem 1.1. If X ⊆ Rd is infinite then there is a 2-rainbow Y ⊆ X with |Y | = |X|. 2010 Mathematics Subject Classification: 03E75, 03C98 Key Words and Phrases: Ramsey theory, distinct volumes, indiscernibles ∗Department of Computer Science, University of Maryland at College Park, College Park, MD 20742. Email: [email protected]. †Department of Mathematics, University of Maryland at College Park, College Park, MD 20742. Email: ds [email protected]. The author was partially supported by NSF Research Grant DMS-1308546. 1 The case when |X| is countable can be dealt with quickly using the canonical Ramsey theorem of Erd˝os and Rado [4]. Alternatively, it is equivalent to apply Ramsey’s theorem to a coloring g : X → c, for c<ω large enough. Namely, given s ∈ X , define g(s) so as to encode the set 2a 2a of all pairs (u, v) from s having the same volume. For our purposes, we find this latter approach a more natural, although some of what we do could be phrased in the language of the canonical Ramsey theorem. Erd˝os’s proof of Theorem 1.1 is complicated by the possibility that |X| is singular. He notes the following holds by an easier proof: Theorem 1.2. If X ⊆ Rd is infinite with |X| regular, and 2 ≤ a ≤ d+1, then there is an a-rainbow Y ⊆ X with |Y | = |X|. Erd˝os also gives the following example: Theorem 1.3. If λ ≤ 2ℵ0 is singular then there is X ⊆ Rd of size λ, such that there is no 3-rainbow Y ⊆ X with |Y | = λ. 2 Proof. Write cof(κ) = λ. Let (ℓα : α < λ) be λ-many parallel lines in R . Let (κα : α < λ) be a cofinal sequence of regular cardinals in κ. Choose Xα ⊆ ℓα of cardinality κα and let X = Xα. Sα<λ Let Y ⊆ X have cardinality κ. We claim that Y cannot be 3-rainbow. Indeed, write Yα = Y ∩Xα = Y ∩ ℓα. Then there must be cofinally many α < λ with Yα infinite, as otherwise |Y |≤ κα + λ < λ for some α < λ. Thus we can find α < β < λ such that Yα and Yβ are both infinite. Let v0, v1 be two distinct points in Yα, and let w0, w1 be two distinct points in Yβ. Then the triangles (v0, v1, w0) and (v0, v1, w1) have the same nonzero area. We are interested in strengthenings and generalizations of Theorem 1.2 for uncountable sets. We will give stronger canonization results than just a-rainbow. Namely, say that X is strongly a-rainbow if all a-subsets of X yield distinct, nonzero volumes, and say that X is strictly a-rainbow if X is strongly a′-rainbow for all a′ ≤ a, and X is a subset of an a − 1-dimensional hyperplane. (In particular, all a + 1-subsets of X have volume 0.) As an example, if an,i : n<ω,i<d are d algebraically independent reals, and if we set an = (an,0, . , an,d−1) ∈ R , then X := {an : n<ω} ′ is strongly d+1-rainbow, and thus strictly d+1-rainbow. Moreover, if ρ : Rd → Rd is any isometric embedding, then the image of X under ρ is also strictly d + 1-rainbow. In Section 2, we begin by reviewing some model-theoretic results of Shelah (Theorems 2.2, 2.3, 2.4), dealing with the following situation: we are given T stable, M |= T , and X ⊆ M infinite, and we try to find Y ⊆ X with |Y | = |X| and Y indiscernible. These theorems only deal with the case where |X| is regular; Theorem 1.3 above shows that obstacles exist for the singular case. The problem is the presence of an equivalence relation E on X that divides X into fewer than κ-many classes, each of size less than κ. Theorems 2.5 and 2.6 together demonstrate that this is the only obstruction, using a weakened notion of indiscernibility with respect to an equivalence relation E. We remark that the combinatorial argument for Theorem 2.6 has other applications; we give a purely finitary analogue in Theorem 2.8. In Section 3, we consider X ⊆ Rd of size κ for some uncountable cardinal κ, and we try to get Y ⊆ X of size κ which is as nice as possible with respect to a-ary volumes for a ≤ d + 1, using the 2 results of Section 2. We prove in Theorem 3.3 that for every regular cardinal κ ≤ 2ℵ0 , and for every X ⊆ Rd of size κ, there is some X′ ⊆ X of size κ and some 2 ≤ a ≤ d + 1 such that X is strictly a-rainbow. We proceed as follows: given X ⊆ Rd of size κ, we obtain a sufficiently indiscernible Y ⊆ X using Theorem 2.4, using the stability of (C, +, ·, 0, 1). Then we apply geometric arguments to argue that Y is strictly a-rainbow for some a. We note that it is possible to prove Theorem 3.3 directly, similarly to Theorem 1.2. For singular cardinals, we know from Theorem 2.6 that there is some finite list of possible con- figurations, although we cannot identify it explictly. We are at least able to give some information about what the configurations look like in Theorem 3.4; in particular, they are all 2-rainbow, and so we recover Erd˝os’s Theorem 1.1. In Section 4, we consider what happens for X ⊆ Rd which is reasonably definable. Our main result is Theorem 4.3: if P ⊆ Rd is perfect, then there is a perfect Q ⊆ P and some a ≤ d +1 such that Q is strictly a-rainbow. Our main tool is a Ramsey-theoretic result of Blass [1] concerning colorings of perfect trees. In Section 5, we show it is independent of ZF whether or not every uncountable subset of R has an uncountable 2-rainbow subset. In this paper we work in ZFC, with the exception of Section 4, which is in ZF + DC. 2 Some remarks on indiscernibles We first review the notion of local indiscernibility, following Shelah [10]. T will always be a complete first order theory in a countable language. Suppose ∆ is a collection of formulas of T , M |= T and A ⊆ M. Given a finite tuple b from <ω M, define tp∆(b/A) to be the set of all formulas φ(x, a) such that a ∈ A and φ(x, y) ∈ ∆ and M |= φ(b, a). d Suppose also that I is an index set, and (ai : i ∈ I) is a sequence from M for some d<ω. Then: • We say that (ai : i ∈ I) is ∆-indiscernible over A if: given i0,...,in−1 all distinct ele- ments of I, and given j0,...,jn−1 also distinct elements from I, then tp∆(ai0 ,..., ain−1 /A)= tp∆(bi0 ,..., bin−1 ). In this case the indexing doesn’t matter and so we also say that {ai : i ∈ I} is indiscernible over A. • If I is linearly ordered, then we say that (ai : i ∈ I) is ∆-order-indiscernible over A if: for every i0 <...<in−1, j0 <...<jn−1 from X, tp∆(ai0 ,..., ain−1 /A)= tp∆(bi0 ,..., bin−1 ). When we do not mention A, we mean A = ∅. The following is an easy application of Ramsey’s theorem (as recorded for instance in Lemma 2.3 of Chapter I of [10]: Theorem 2.1. Suppose T is a complete first order theory in a countable language, and ∆ is a finite collection of formulas of T .
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